Diffusion-driven instability and bifurcation in the Lengyel-Epstein system. (English) Zbl 1146.35384

Summary: Lengyel-Epstein reaction-diffusion system of the CIMA reaction is considered. We derive the precise conditions on the parameters so that the spatial homogeneous equilibrium solution and the spatial homogeneous periodic solution become Turing unstable or diffusively unstable. We also perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.


35K57 Reaction-diffusion equations
35B10 Periodic solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92B05 General biology and biomathematics
35B32 Bifurcations in context of PDEs
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[1] Casten, R.G.; Holland, C.J., Stability properties of solutions to systems of reaction – diffusion equations, SIAM J. appl. math., 33, 353-364, (1977) · Zbl 0372.35044
[2] Crandall, M.G.; Rabinowitz, P.H., The Hopf bifurcation theorem in infinite dimensions, Arch. rat. mech. anal., 67, 1, 53-72, (1977) · Zbl 0385.34020
[3] De Kepper, P.; Castets, V.; Dulos, E.; Boissonade, J., Turing-type chemical patterns in the chlorite – iodide – malonic acid reaction, Physica D, 49, 161-169, (1991)
[4] Epstein, I.R.; Pojman, J.A., An introduction to nonlinear chemical dynamics, (1998), Oxford University Press Oxford
[5] Hassard, B.D.; Kazarinoff, N.D.; Wan, Y.-H., Theory and application of Hopf bifurcation, (1981), Cambridge University Press Cambridge, MA · Zbl 0474.34002
[6] Jang, J.; Ni, W.M.; Tang, M., Global bifurcation and structure of Turing patterns in the 1-D lengyel – epstein model, J. dynam. differential equations, 16, 2, 297-320, (2005) · Zbl 1072.35091
[7] Lengyel, I.; Epstein, I.R., Modeling of Turing structure in the chlorite – iodide – malonic acid – starch reaction system, Science, 251, 650-652, (1991)
[8] Lengyel, I.; Epstein, I.R., A chemical approach to designing Turing patterns in reaction – diffusion system, Proc. natl. acad. sci. USA, 89, 3977-3979, (1992) · Zbl 0745.92002
[9] J.D. Murray, Mathematical Biology, third ed., I. An introduction. Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002; II. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics, vol. 18, Springer, New York, 2003.
[10] Ni, W.; Tang, M., Turing patterns in the lengyel – epstein system for the CIMA reaction, Tran. am. math. soc., 357, 3953-3969, (2005) · Zbl 1074.35051
[11] Rovinsky, A.; Menzinger, M., Interaction of Turing and Hopf bifurcations in chemical systems, Phys. rev. A (3), 46, 10, 6315-6322, (1992)
[12] Ruan, S., Diffusion-driven instability in the gierer – meinhardt model of morphogenesis, Natural resource modeling, 11, 131-142, (1998)
[13] Turing, A.M., The chemical basis of morphogenesis, Phil. trans. R. soc. London ser. B, 237, 37-72, (1952) · Zbl 1403.92034
[14] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, (1991), Springer New York
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